Lensless imaging with reduced aperture

ABSTRACT

An image of an object can be synthesized either from the Fourier components of the electric field or from the Fourier components of the intensity distribution. Imaging with a lens is equivalent to assembling the Fourier components of the electric field in the image plane. This invention provides a method and a means for lensless imaging by assembling the Fourier components of the intensity distribution and combining them to form the image with the use of amplitude splitting interferometer. The angular spectrum of the electromagnetic radiation consists of wavefronts propagating at different angles. The amplitude of each wavefront is split and interfered with itself to create sinusoidal fringe patterns having different spatial frequencies. The sinusoidal fringe patterns are combined to form an image of the object. This method applies to coherent and incoherent light.

RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.11/331,400 filed on Jan. 12, 2006 which claims priority under 35 U.S.C.§119(e) to U.S. Provisional Application Ser. No. 60/643,327 entitled“Interferometric Imaging With Reduced Aperture,” filed on Jan. 12, 2005,which is herein incorporated by reference in its entirety.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention concerns enhancing the resolution of optical imagingsystems. This invention relates to imaging interferometers that createan image of a laterally extended object.

2. Discussion of Related Art

There are two kinds of imaging systems, those that form an imagedirectly and those that synthesize the image from its Fouriercomponents. When an object is illuminated with electromagnetic radiationit creates a diffraction pattern. If the radiation is spatiallycoherent, then the diffraction consists of a summation of plane wavesdiverging away from the object each representing one spatial frequencycomponent of the electric field distribution on the object. The angle ofdiffraction of a particular wave is equal to the spatial frequency timesthe wavelength of light. Each one of these waves has amplitude and phaseassociated with it. On a screen in the far field, the diffractionproduces an irradiance pattern, which is identical to the magnitudesquared of the Fourier Transform of the electric field leaving theobject. The phase information is lost. On the other hand if the body isilluminated with incoherent radiation or if it is self luminous, such asdistant stars, then the illumination is uniform but the degree ofcoherence in the aperture plane is given by the Fourier Transform of theirradiance distribution on the object, according to the vanCittert-Zemike theorem, which is used extensively in astronomy.

Conventional imaging systems, such as a lens or holography, aim atcapturing and recombining most of the waves that are diffracted from theobject in order to construct the image. In principle, if all thedivergent waves are redirected to intersect and overlap over some regionof space while maintaining their original amplitudes, relative phasesand angles then they would recreate a perfect image of the object. Thefailure of such systems to reproduce a perfect image is mainly due tothe inability to capture the high frequencies, besides imperfections inthe quality of the optical surfaces. In fact, image formation is aninterferometric process. A lens is capable of forming an image of acoherent as well as an incoherent object because it has the propertythat all the rays and waves travel equal optical paths between theobject plane and image plane. Thus, it does not matter what thecoherence length is, the lens will form an image on-axis because theoptical path length difference there is zero. The interference takesplace over the entire image space and not just on-axis. At off-axispoints in the image plane the path length difference changes gradually.Thus, the ability to interfere completely off-axis depends on the degreeof coherence among the waves. The absence of the high frequency waves infinite aperture imaging systems creates artifacts in the image plane,which account for the degradation of the performance of coherent imagingsystems. By contrast, the lack of coherence smoothes out the off-axisintensity variations in incoherent systems.

It is desired to broaden the aperture of imaging systems in order toenhance the imaging resolution. It is also desired to capture morephotons from dim or rapidly varying sources to improve the signal tonoise ratio. An increase in the size of the aperture of a lens orprimary parabolic mirror is impractical beyond a certain limit becausethe cost of fabrication of large focusing elements to optical tolerancesbecomes prohibitive. Holographic imaging techniques replace the lenswith a holographically fabricated grating. The phase of the firstdiffracted order from the holographic grating varies as the quadratic ofthe distance off-axis. This is analogous to the phase incurred by a raytraversing a lens, which varies quadratically with the radial positionof the ray. A holographically fabricated grating preserves the phase andangle relationships among the diffracted waves similar to alens, whichallows it to reproduce the image with fidelity during reconstruction. Aholographic grating accomplishes by diffraction what a lens does byrefraction.

Conventional imaging systems utilize a focusing element, such as lens orparabolic mirror at full aperture to image distant objects in the focalplane. The focused image is susceptible to atmospheric aberrations andto imperfections of the optical surface. In order to reduce the effectof aberrations, it is desired to defocus or spread the light in theimage plane. This can be accomplished by restricting the aperture, i.e.the use of synthetic aperture techniques and non-focusing opticalelements, such as planar mirrors.

Coherent Imaging

The imaging of a coherent object can be achieved using a conventionalfull-aperture system, such as a lens or hologram. The electric fields ofthe diffracted waves add in amplitude and phase when recombined in theimage plane. This produces a replica of the object if the amplitude,phase and angle relationships among the waves are preserved. It is notnecessary in conventional imaging systems to measure or know themagnitude or phase of the electric fields. As long as the amplitude,phase and angle relationships, corresponding to the magnification of theimaging system, are not perturbed then the optical system willreconstruct the object with high fidelity. The burden of conventionalsystems is capturing the high frequency components of the diffractedwaves to achieve a more complete interference. This requires biggerapertures, which increase the cost of the system significantly. Coherentimaging systems used in photolithography for the fabrication ofelectronic circuits aim to achieve sub-wavelength resolution.

Synthetic Aperture

Unconventional systems attempt to create the effect of a large aperturesynthetically by sampling a subset of the diffracted waves with the useof two or more sub-apertures. Since imaging along two axes is usuallydesired and to limit the displacement of each sub-aperture, severalsparsely-located sub-apertures are deployed in the pupil plane. Thebasic technique entails the use of two very narrow apertures, e.g.pinholes as in Young's experiment, so that the diffraction effectsbecome dominant. If the pinholes are placed in the far field of acoherently illuminated object, then each pin hole intercepts only onediffracted wave from the object. If the object is placed on the opticalaxis then the two pinholes capture the conjugate positive and negativefrequencies of its Fourier Transform, i.e. the waves travelingsymmetrically off-axis. The goal is to measure the amplitude and phaseof the spatial frequency components of the field by mapping its FourierTransform in the frequency domain. The diffraction from the pinholescreates sinusoidal fringes in the image plane. The spatial frequency ofthe fringes is determined by the spacing between the pinholes. Thevisibility of the fringes depends on the ratio of the amplitudes of thetwo interfering waves and the phase of the fringes represents thedifference between the phases of the Fourier components. Unequalamplitudes cause a decrease in visibility or modulation of the fringes.A phase difference between the waves causes a shift of the centralfringe off-axis. By varying the spacing between the holes the entirespatial spectrum of the object can be measured. It is worth noting thatfor a coherently illuminated object the Fourier Transform is notnecessarily an even function, i.e. the magnitudes and phases of theconjugate positive and negative frequencies can differ. The object issubsequently reconstructed in the spatial domain by inverse Fouriertransforming the data. Thus, synthetic aperture techniques inevitablyinvolve computation in a two step process. Similarly, holographyperforms image reconstruction is a two step process, namely the writeand read cycles. For this reason holography is not considered to be areal-time process. Nevertheless, the processing times can be shortened.This adds delay in the processing of the image, which becomes a concernespecially for moving targets.

Incoherent Imaging

In the case of an incoherently illuminated or a self luminous object themagnitude and phase of the electric field cannot be uniquely defined.The irradiance in the aperture plane is uniform and the coherencefunction is given by the Fourier Transform of the irradiance of theobject according to the van Cittert-Zemike theorem. Thus, thedegradation in the visibility of the fringes is due to the lack ofperfect coherence due to the diffraction from the pinholes of all theincident waves. The complex visibility of the central fringe, which ismeasured experimentally, is equal to the coherence function. Theenvelope of the fringes decays due to the finite coherence length of thelight source and due to diffraction from the finite-width apertures.Thus, by varying the spacing between the apertures the entire spatialspectrum of the irradiance of the object can be mapped in the frequencydomain, which is then inverted to obtain the irradiance distribution inthe spatial domain, i.e. the image. Synthetic aperture techniques areknown as Fourier imaging because the Fourier Transform of the image,rather than the image itself is obtained, which requires furthercomputation to derive the image. By contrast, the full aperture lens andhologram are direct imaging techniques because the overlapping wavesconstruct the image directly.

In either coherent or incoherent imaging it would be necessary toincrease the size of the aperture in order to improve the imagingresolution. Since a Fourier Transform relationship exists between theobject plane and the pupil plane, the resolution in one plane isinversely proportional to the total sampling interval in the other. Forthis reason, the apertures of the very large baseline telescope (VLBT)are pushed as far apart as possible in order to achieve nano-radianresolution. For example, it is desired to utilize apertures withdiameters on the order of 10 to 30 meters to image space andastronomical objects to achieve an angular resolution of 10 to 50nano-radians in the visible. It would be impractical and prohibitivelyexpensive to construct a curved mirror or lens of this diameter out of amonolithic piece of glass while maintaining a high quality opticalsurface. The advantage of the synthetic aperture technique is that itachieves the resolution of a very large aperture with two smallerapertures positioned diametrically opposite each other.

Lensless Imaging

Imaging systems can be classified either as direct or Fourier. The lensis the only optical device that can produce a direct two-dimensionalimage instantaneously. Holography produces a direct image but it is atwo-step process; so is synthetic aperture. However, synthetic aperturecan extend the aperture beyond the limits of a lensed system. For thisreason, it has been the goal of imaging system designers to eliminatethe lens, especially in the push toward bigger apertures. It is worthnoting, however, that even though holography replaces the lens with agrating, the synthetic aperture technique does not preclude using alens. A lens can be masked entirely except for two pinholes, forexample, and the interference pattern is transformed from a focused Airypattern to a sinusoidal interferogram. In fact, A. A. Michelson's earlyexperiment in 1920 atop Mount Wilson, which gave birth to stellarinterferometry, consisted of a lensed synthetic aperture system. Hecovered most of the 10′ telescope except for two 6″ diameter holes.However, Michelson did not produce a complete imaging system. His goalwas to measure the stellar diameter. He managed to observe the fringesand make quantitative measurements in spite of atmospheric turbulence,which caused the fringes to wander and drift. This demonstrated thetolerance of the synthetic aperture technique to atmosphericdisturbances by virtue of spreading the light in the image plane, i.e.observing an interferogram instead of a focused image, besides theability to position the outer mirrors at distances greater than thediameter of the telescope. Nevertheless, the potential of the syntheticaperture technique is to deliver high resolution imaging without using alens.

Therefore, the goal is to produce a two-dimensional image anddemonstrate high resolution without using a lens. Another goal is toproduce as close to a direct image as possible, i.e. to display theimage in real space and time by simplifying the algorithm and minimizingthe computations.

Magnification

A fundamental aspect of imaging is determining the correctmagnification. The overlap of waves in the image plane of a lens forms aperfect image, but the observed image cannot be related to the realobject unless the distance between the object and the lens is known. Inprinciple, the plane of the image and the magnification can bedetermined experimentally at the location of best focus. However, fordistant objects the image plane coincides with the focal plane, and thelinear magnification vanishes. If no focusing element is used then thefringes become non-localized and form in any region of overlap in thefar field. The lens reproduces the far field or Fraunhofer conditions inits focal plane and provides a length scale, i.e. focal length by whichoff-axis distances are measured. Holography, on the other hand, does notfocus or localize the fringes and usually produces an image with unitmagnification if the same wavelength that was used to write the hologramis used again to read it. Holography circumvents the issue ofmagnification; however the real image is formed symmetrically about theplane of the hologram at equal distance from where the object originallystood. Thus, it would be impractical to image distant or space objectsholographically because the image would be located far away in space inthe opposite direction from the object. The synthetic aperture techniquedoes not resolve the issue of magnification either without priorknowledge of the size of the object or its distance. Under coherentillumination the pin holes sample two conjugate frequencies from thespectrum of the electric field. The measurement of the complexvisibility yields the ratio of the two amplitudes of the waves and theirphase difference. But it is not possible to relate the measurement to aspecific spatial frequency component or angle of diffraction unless thedistance to the object is known. In the absence of such knowledge, animage resembling the object can be synthesized by inverse Fouriertransformation but it would be related to the real object through anunknown scaling factor. The spatial frequency of the fringes in theimage plane is not related to the spatial frequencies of the object. Itdepends on the spacing between the holes and the distance between thepupil and image planes. In the coherent case it is not possible toidentify the central fringe because all the fringes have equalvisibility, and therefore it is not possible to determine the exactmagnification. In the case of incoherent illumination the measurement ofvisibility yields an estimate of the complex degree of coherence. It ispossible to pin point the central fringe in an incoherent interferogrambecause of the decaying envelope of the fringes due to finite temporalcoherence. However, this does not yield a direct measurement of thespatial components of the irradiance function of the object. Either thesize of the object or its distance from the pupil plane must be known inorder to properly characterize the object. However, for very distantobjects the visibility measurement can be related to the angle subtendedby the diameter of the object as observed from a point in the pupilplane. This fact is widely used in astronomy to estimate angulardiameters of stars, rather than image objects at finite distances withknown magnification.

Determining the scaling factor between the reconstructed image and theactual object requires knowledge of either distance to the object or tothe image plane. However, in an afocal system, which does not use a lensthere is no unique image plane. The fringes are non-localized and can beobserved at any plane within the overlap region in the far field.

The synthetic aperture technique does not measure the phase differencebetween two non-conjugate orders. The inversion of the Fourier Transformnecessary to reconstruct the image requires knowledge of the phaserelationships among all the constituent orders. Thus, the syntheticaperture technique does not measure the amplitude and phase as afunction of the angle of diffraction and second, measurement of thecomplex visibility does not yield enough phase relationships among thediffracted orders to permit computation of the inverse FourierTransform.

Longitudinal and Lateral Imaging

It is desired to enhance the resolution of optical imaging systems. Anobject has three-dimensions, one longitudinal along the optical axis,and two lateral dimensions. It is desired to enhance the longitudinal aswell as lateral resolutions. Light from an idealized point source passesthrough an optical system and is projected on a screen in the far fieldperpendicular to the optical axis. The light interferes on the screenand forms the image of the point source, which is the point spreadfunction (PSF). Information about the point source is obtained byanalyzing the PSF. Wave front dividing systems consist of one or moreapertures. The PSF is the Fourier Transform of the aperture plane.Qualitatively described, the width of the light distribution in the farfield is inversely proportional to the width of the apertures. The PSFof a single circular aperture is the Airy pattern, while that of arectangular aperture is the sin(x)/x function. A typical PSF of a singleaperture system has a width equal to ƒλ/α, where α and ƒ are the widthand focal length of the imaging system, respectively, and λ is thewavelength of light. The PSF of a system of two apertures separated by adistance D consists of sinusoidal fringes of period ƒλ/D. The fringesare modulated by an envelope of width ƒλ/α, corresponding to the width aof each aperture. Thus, the PSF of a two-aperture system consists of D/αfringes. This is typical of diffraction-limited systems. Another factor,which affects the interference, is the degree of coherence of the lightsource. A perfectly temporally coherent point source produces fringes,which are only diffraction limited. A partially coherent point sourceyields a number of fringes, which is equal to the coherence length ofthe source, L_(o), divided by the wavelength λ. Thus, the number offringes is given by D/α, or L_(o)/λ, whichever is smaller; that is thesystem is either diffraction limited or coherence limited.

Wavefront Division and Amplitude Division

Wave front division (WD) imaging systems, such as the Michelson stellarinterferometer, which is based on Young's Experiment, focus the light ontwo narrow apertures, such as slits or holes. Several apertures orsub-apertures can be used, which create a speckle pattern in the focalplane. These systems are diffraction limited. Amplitude dividing (AD)systems, such as the Mach-Zehnder or Michelson interferometer are oftenused to create sinusoidal fringes by interfering two collimated beams atfull aperture. These systems are usually coherence limited rather thandiffraction limited. It is often desired to image faint sources. Forthis reason the beams are usually partially focused or compressed to anarrower cross-section using optical reduction. This enhances the signalto noise ratio and improves the quality of the image. The intersectingbeams subtend a half angle θ. For small angles, the period of thefringes is equal to λ/2θ or λ/(2 NA), where NA is the numerical apertureof the imaging system. The position of the fringes is determined by thephase difference between the beams. A shift in phase causes the fringesto move in the observation plane. The key to enhancing the resolution ofimaging systems is tracking the motion of the fringes. Longitudinal andlateral displacements of the point source cause the fringes to move bydifferent amounts. For example, WD systems are less sensitive tolongitudinal motion because the phase difference at the two aperturesremains almost unchanged. Similarly, AD systems are less sensitive tolateral motion of the mirror. For this reason, WD systems, such as thestellar interferometer and the microscope are used for lateral imaging,while AD systems, such as the Michelson and Mach Zehnder andTwyman-Green interferometers are used for longitudinal imaging with theuse of a reference arm. Longitudinal imaging systems usually image asingle source, whereas lateral imaging systems image multi sources.

Image of a Point Source Through Two Apertures

The image of a point source through a system of two apertures is asinusoidal fringe pattern, which has a limited lateral range. The numberof fringes is determined by the diffraction from the two apertures andthe temporal coherence properties of the light source. The period of thefringes is determined by the numerical aperture of the optical systemand the phase is determined by the angular position of the point sourcerelative to the optical axis. The amplitude of the fringes is determinedby the brightness of the source. The central fringe corresponds to thepoint on the observation plane where the optical path length differencebetween the two arms of the interferometer is zero. The images of twopoint sources overlap. The phase difference between the two centralfringes of two point sources is the image of the object containedbetween those two points. The images of several points yield overlappingsinusoidal fringes with different amplitudes and phases. The lightdistribution consisting of the superposition of all these fringes isrecorded with a CCD array or a photodetector. Imaging of the points isequivalent to unraveling the individual phases and amplitudes.

Visibility

The light source is often faint and very distant from the opticalsystem. The light must travel through some atmosphere before it reachesthe detector. The atmosphere often interferes with and degrades thequality of the image. Specifically, the atmosphere introduces a randomand time varying phase shift to the fringes. However, the relative phaseshifts between the fringes and the period of the fringes are unaffected.An effective technique to cancel the effect of the drift and recover theoriginal phases is to measure the visibility of the fringes, i.e. theratio of intensity variation to the average optical power. Thevisibility of the superposition of fringes is a function of the relativephases but also depends on the amplitudes. For this reason, visibilitytechniques are usually limited to bright sources of equal brightness.

Synthetic Imaging Technique

Two-aperture systems have the potential for higher angular resolutionthan single aperture systems of comparable size because of the abilityto discern a phase change of a fraction of 2π. A typical interferometricconfiguration that is commonly used for lateral imaging is the Michelsonstellar interferometer (MSI), which is based on Young's Experiment. Aschematic representation of this configuration is shown in FIG. 1. Thebasic concept consists of only two narrow slits, however, mirrors areadded to gain sensitivity because the mirrors can be placed much fartherapart. Collimated light from distant sources is focused by the system ofmirrors M₁ and M₂ on slits S₁ and S₂ by mirrors M₃ and M₄. The opticalpath lengths traveled by all the rays in one arm of the interferometerare equal. For a point source located at an angle θ relative to theaxis, the optical path length difference between the two arms is Lθ,where L is the separation between the outer mirrors M₁ and M₂. Thepurpose of the inner mirrors is to focus the light on the slits. Thedistance between the slits is chosen to yield the desired numericalaperture of the imaging system. The angle between the interfering raysat any point in the image plane is determined by the numerical apertureNA and is independent of the angle of incidence. The period of thefringes, λ/(2 NA), can be chosen arbitrarily as long as the fringes canbe viewed with available photodetectors. The size of the image, i.e. thephase difference between the images of two points is proportional to theseparation L. As L is increased the size of the image increases and thevisibility

of the central fringe changes.

is plotted vs L. The image is obtained by taking the inverse FourierTransform of

(L) according to the Van Cittert-Zemike theorem. Thus, the imagingresolution is inversely proportional to the sampling interval L. Forthis reason the mirrors of the Very Long Baseline stellar Interferometerare placed very far apart, up to hundreds of meters, in order to gainangular resolution. This is equivalent to unraveling the phases of thesinusoidal fringes in the image plane. If it is desired to image N pointsources each having a different brightness, then we have 2N unknownscorresponding to the N phases and N amplitudes. Scanning the mirrors,i.e. changing L is equivalent to providing a set of 2N equations for the2N unknowns. Thus, Fourier transforming

(L) by the Van Cittert-Zemike theorem is equivalent to solving a systemof 2N equations by 2N unknowns. Obtaining the relative phases andamplitudes of the fringes corresponding to the point sources that makeup an object is tantamount to imaging the object. This is known assynthetic imaging.

Sheared Wavefronts

An interferometer splits an original wavefront and then recombines itwith itself. Typically the wavefront is sheared, that is a portion ofthe wavefront interferes with another portion of the same wavefront uponrecombination. Different interferometric configurations exhibitdifferent amounts of shear. For example, wavefront dividinginterferometers exhibit no shear because the original wavefront isdestroyed upon focusing. A new wavefront emerges from each slit and theshear is lost. All the rays being focused on one slit in one arm of theinterferometer have equal optical path lengths. The interference beyondthe plane of the apertures is governed by the diffraction from theslits. By contrast, amplitude splitting interferometers exhibit shear.

Erect and Inverted Shear

The shearing properties of interferometers are best illustrated usingcollimated light. Imagine that light from a point source is collimatedand directed at an amplitude splitting interferometer, such as Michelsonor Mach Zehnder. The incident wavefront is split by a partiallyreflecting/transmitting mirror along two different paths. The splitbeams are redirected by mirrors tilted at the proper angle to createeither spatial or temporal fringes. The fringes are created at fullaperture and the interference is more coherence than diffractionlimited. If an imaginary line is drawn along the middle of the incidentwavefront bisecting it in half, and the two split wavefronts are made tooverlap completely on the detector, then depending on the number ofmirrors encountered in each path each half of the original wavefrontwill either interfere with itself or with the other half uponrecombination. If the difference between the number of reflections alongthe two paths is even, such as the Michelson configuration where eachbeam experiences two reflections, then each half will interfere withitself and the shear is erect. If on the other hand the difference isodd, such as the Mach Zehnder configuration (leaving out the exit beamcombiner) where one beam encounters two mirrors along its path and theother beam only one, then each half of the original wavefront willinterfere with the other half and the shear is inverted. Thisterminology is analogous to the imaging properties of a lens, whichdepending on the location of the object will either create an erect orinverted image. It is usually desirable to achieve complete overlap ofthe two wavefronts. However, the amount of shear can be varied bydisplacing the mirrors to vary the overlap between the two intersectingbeams. In the case of erect shear the shear is constant across thewavefront, i.e. the separation between any two interfering rays in theoriginal wavefront is constant. This is the case of shearinginterferometers, which are commonly used to measure tilt of wavefronts.In an inverted shear configuration the shear varies across thewavefront. An arbitrary number of mirrors can be added to each path ofthe interferometer, which will not change its shearing properties aslong as the difference remains either even or odd.

Diffractionless Interferometer

If the slits in the MSI design of FIG. 1 were removed in order toeliminate their diffraction effects and the two wavefronts were allowedto propagate and interfere at full aperture on the image plane, then wewould obtain the erect shear configuration of FIG. 2. The beams couldalso be partially focused or reduced optically in order to limit thearea of interference and raise the signal to noise ratio. Mirrors M₃ andM₄ are oriented slightly differently from those in FIG. 1 in order tocause the two beams to interfere at a shallow angle, for example about0.75 degree to yield detectable fringes with a period of about 20microns for visible light. The distance between mirrors M₃ and M₄ ischosen in conjunction with the distance to the image plane to yield thedesired NA. The period of the fringes is not critical because thevisibility is measured, which is independent of period. A change of θ inthe angle of incidence corresponding to two point sources yields anoptical path length difference of Lθ, which causes a phase shift of(Lθ/λ)2π regardless of the fringe period. Thus, the phase and image varylinearly with the size of the object. Even though the angle of each beamfalling on the observation plane changes with the angle of incidence,the subtended angle between the two beams remains constant because thebeams track each other. This insures that the period of the fringesremains constant. However, the fringes turn and change their phase inthe observation plane. The result is that two distant point sourcesyield identical fringe patterns that are displaced from each other by aphase shift corresponding to the image. This is similar to the stellarinterferometer of FIG. 1. The imaging resolution is obtained from theanalysis of the visibility V of the central fringe plotted against theouter mirror separation L. For higher resolution, L is increased. Inconclusion, if diffraction plays a significant role in the interferenceor if the shear is erect, then the imaging is obtained from analysis ofthe visibility and the angular resolution is enhanced by increasing thedistance between the outer mirrors.

SUMMARY OF INVENTION

This invention enhances the resolution of optical imaging systems bycreating a phase change due to change in the angle of incidence withoutlateral shear. A beam of light is split and the two beams are recombinedwith complete overlap. The phase change between the two beams varieswith the angle of incidence. This invention concerns creating an imageof a laterally extended object interferometrically without using a lens.

BRIEF DESCRIPTION OF DRAWINGS

The accompanying drawings are not intended to be drawn to scale. In thedrawings, each identical or nearly identical component that isillustrated in various figures is represented by a like numeral. Forpurposes of clarity, not every component may be labeled in everydrawing. In the drawings:

FIG. 1 illustrates a Michelson stellar interferometer;

FIG. 2 illustrates an erect shear interferometer;

FIG. 3 illustrates an inverted shear interferometer;

FIG. 4 illustrates an interferometer using two curved mirrors;

FIG. 5 illustrates an interferometer with reduced cross section;

FIG. 6 illustrates an amplitude splitting imaging interferometer.

DETAILED DESCRIPTION

This invention is not limited in its application to the details ofconstruction and the arrangement of components set forth in thefollowing description or illustrated in the drawings. The invention iscapable of other embodiments and of being practiced or of being carriedout in various ways. Also, the phraseology and terminology used hereinis for the purpose of description and should not be regarded aslimiting. The use of “including,” “comprising,” or “having,”“containing”, “involving”, and variations thereof herein, is meant toencompass the items listed thereafter and equivalents thereof as well asadditional items.

Motion of the Fringes

The distinction between erect and inverted shear has major implicationson the motion of the fringes. In the case of erect shear, the fringes donot change their period as the angle of incidence changes. They merelychange phase and displace, similar to divided wavefront diffractionlimited systems. On the other hand, in inverted shear systems thefringes change their period in response to angular variations. Sinceimaging is a matter of tracking and discerning the motion of thefringes, the two configurations have distinct imaging properties. Thiscan be achieved simply by adding one mirror in the path of one arm ofthe interferometer to make it asymmetrical.

Fringe Detection

The interference pattern from two apertures consists of spatial fringesand is usually detected with the use of a CCD array. Alternatively, thefringes can be scanned past a fixed CCD and recorded sequentially intime. The interfering beams can also be combined with the use of apartially reflecting/transmitting mirror and detected with a simplephotodetector to produce temporal fringes. Similarly, the interferencepattern of an AD system can be observed spatially by orienting the beamsto intersect at a shallow angle, such as 1.5 degree. This yields afringe period of about 20 microns for visible light, which covers twopixels of a typical state-of-the-art CCD array. The emerging wave frontsin AD systems could also be made parallel and collinear and the fringesobserved temporally. Since the period of a fringe, λ/2θ, depends on theangle of incidence θ, different wave fronts set up fringe patterns withdifferent spatial frequencies. Proper sampling of the spatial lightdistribution requires use of CCD arrays whose pixel size is shorter thanhalf the period of the highest spatial frequency, according to theNyquist criterion. Also, the sampling interval and the number of pixels,i.e. the length of the CCD array is inversely proportional to thespatial frequency resolution. Thus, achieving a certain frequency, henceangular resolution requires the use of a CCD with a certain minimumlength.

Frequency Technique

If one mirror, denoted by M5 in FIG. 3, is added to one arm of theinterferometer, then a condition of inverted shear is created. Theperiod of the fringes changes with the angle of incidence because thetwo interfering beams turn in opposite directions. Thus, two pointsources create two fringe systems with unequal periods. Not only arethey displaced in phase but also they have different spatialfrequencies. This gives rise to a variable phase difference. The imagingproperties of this system can be obtained by taking the discrete FourierTransform (DFT) of the fringes themselves, rather than scanning themirrors and Fourier transforming the Visibility function as with erectshear. The visibility cannot be defined properly because the constituentperiodic waves have different spatial frequencies. If the rays weretraced along each path of the interferometer then it is found that theseparation between the two rays interfering at any point varies acrossthe observation plane, hence inverted shear. In this case the actualperiod of the fringes does matter because the spatial frequency ismeasured. The phase and image vary non-linearly with the angle ofincidence because the constant of proportionality, L, itself varies withθ. It is this property that allows the achievement of a high angularresolution for smaller L. Therefore; it becomes critical to know theabsolute fringe period because it is the spatial frequency that isactually measured rather than the visibility. The angular resolutionbecomes a matter of discerning frequency rather than phase changes. Thefrequency resolution ƒ_(res) is directly proportional to the angularresolution, ƒ_(res)=2θ_(res)/λ. Therefore; a reading of the DFT of thefringe pattern yields a direct reading of the angular position andbrightness of the point sources. This is an alternative syntheticimaging technique, which is analogous to Fourier transforming

(L) by the Van Cittert-Zermike theorem.

Frequency Resolution

The frequency resolution obtained through a discrete Fouriertransformation of a spatial signal is inversely proportional to thesampling interval; in this case the length of the fringe pattern L_(fr).This implies that L_(fr) must be equal to L/2 in order for the frequencymeasuring technique to yield the same angular resolution as the phasemeasuring technique. The fringes must extend over a length half theseparation of the mirrors. This suggests that the frequency technique ismore applicable to temporally coherent systems. This states that inorder to achieve a certain angular resolution, the optical instrumentmust have a certain minimum size, which proves that the frequencytechnique does not violate the diffraction limit, even though it mayutilize an optical system, which is not diffraction limited. Thus, thereare two alternative techniques for synthetic imaging and angularresolution, the phase technique, which was demonstrated by Michelsonover a century ago, and the frequency technique, which is described inthis paper. Each technique has its advantages and drawbacks. Measuringthe phase difference between two fringes requires only a few cycles andcan be done with broadband or incoherent sources, such as starlight. Theangular resolution increases with the separation between the mirrors; L.Measuring frequency does not depend on L, but requires fairly coherentor single frequency sources. Frequency measurement is equivalent tocounting an integer number of cycles. If it is desired to resolve afrequency difference Δf between two signals whose frequencies are ƒ and(ƒ-Δƒ), then the signals must be sampled for ƒ/Δƒ cycles over aninterval 1/Δƒ equivalent to the beat length. Nevertheless, the frequencytechnique has the advantage of being more robust and accurate than thevisibility technique, which depends on amplitude, because themeasurement of frequency is unaffected by intensity fluctuations.Another advantage of the frequency technique is that the CCD array canbe segmented since the phase is not being tracked. This obviates theneed for long monolithic CCD arrays. It is possible to magnify theangular change and effectively reduce the beat length with the use ofoptical reduction techniques or partial focusing, which also enhancesthe signal to noise ratio.

Scanning the Angle of Incidence to Vary the Size of the Image

The ideal solution would be a combination of both techniques to maintaina good angular resolution while reducing L, by making use of the factthat the phase changes non-linearly with the angle of incidence. Thepurpose of increasing the distance between the mirrors in thediffraction limited configuration was to increase the phasecorresponding to a certain angular separation θ, because the opticalpath length is equal to Lθ. The same objective can be achieved with theinverted shear configuration by scanning the angle of incidence withoutchanging L, because the size of the image varies with the angle ofincidence. Thus, it is possible to increase the phase difference betweenthe fringes corresponding to two point sources simply by turning theinterferometer past the stationary point sources so as to change theirangular positions relative to the axis. The phase is very sensitive tominute changes in the angle near normal incidence as the period of thefringes becomes infinite. This technique can be used to vary therelative phases of the sinusoidal fringes by turning the interferometerincrementally to yield a system of 2N equations for 2N unknowns, whichcan be solved for the N phases and N amplitudes, effectively producing asynthetic image of N points analogous to the visibility technique.Scanning is necessary for fairly coherent sources in order to capturethe entire fringe pattern with a limited size CCD array. As theinterferometer is turned, the fringes corresponding to different pointsources move past the photodetector while changing their relativephases. The CCD records the different frames and produces a temporalsignal from which the different phases can be computed. If it is desiredto discern two frequency components that are ƒ and (ƒ-Δƒ), then theangle is scanned until the accumulated phase difference between the twofringes passing over the active region of the photodetector is acomplete cycle.

The two interfering beams in FIG. 3 are incident along two differentpaths A and B, and intercepted by mirrors M₁ and M₂. Mirrors M₃ and M₄direct the light to interfere at the proper angle in the observationplane. The beams A and B originate from the same point source, which isassumed to be distant so that the light reaching the interferometer iscollimated and the two beams are separated by a distance L. Achieving acertain angular resolution requires a certain minimum separation. Thisinvention teaches a method of reducing L while maintaining a goodangular resolution.

DESCRIPTION OF THE INVENTION

This invention is described with the use of geometric optics as shown inFIG. 4. The two beams A1 and B1 are derived from the same wavefront C.Incident beam C is split by a system of two parallel mirrors M₆ and M₇.Beam C is incident on mirror M₆ at angle θ₁. The two parallel beams A₁and B₁ are separated by a distance L₁ and intercepted by a curved mirrorCM₁, whose axis E₁ is parallel to the direction of A₁ and B₁. Thedistance between beam A₁ and axis E₁ of CM₁ is denoted by X₁. The curvedmirror CM₁ focuses the beams A₁ and B₁ at its focus F. The focused beamsA₁ and B₁ are further intercepted by a second curved mirror CM₂, whichis confocal, i.e. shares the same focal point F with the first curvedmirror but whose axis E₂ is tilted by an angle θ_(o) relative to axis E₁of CM₁. The angle θ_(o) can take any value between −π and π. The curvedmirrors have two focal lengths ƒ₁ and ƒ₂ that are not necessarily equaland can have any suitable shape that provides a focus, such as forexample parabolic or paraboloidal. Upon reflection from CM₂ the twobeams A₂ and B₂ are again parallel and separated by a distance L₂. Theyare recombined at D using a system of two parallel mirrors M₈ and M₉.Beam A₂ is incident on mirror M₈ at angle θ₂. The mirrors M₈ and M₉ areoriented so that a change in θ₁ causes a change in θ₂ of the samepolarity as in FIG. 4. Alternatively, the mirrors M₈ and M₉ can beflipped symmetrically about the A₂ direction to yield the oppositepolarity. The distances L₁ and L₂ are not necessarily equal. Similarly,the angles θ₁ and θ₂ are not necessarily equal. A change dθ₁, in theangle θ₁ causes the phase of beam A₁ to advance by (L₁dθ₁/λ)2π relativeto that of beam B₁. It also causes the angle θ₂ to change by dθ₂ and thephase of beam A₂ to retard by (L₂dθ₂/λ)2π relative to that of beam B₂.Thus, the total optical path length change is L₁dθ₁-L₂dθ₂ uponrecombination at D. The different parameters can be chosen so that L₁dθ₁is different from L₂dθ₂ when θ_(o) is different from zero or π. Thisyields a net phase change in response to a change in the angle ofincidence.

Example 1

As an example the parameters ƒ₁, ƒ₂, X₁, and L₁ are chosen as follows:ƒ₁=ƒ₂=ƒ, X₁=0.828ƒ, and L₁=0.414ƒ, where ƒ is an arbitrary number, whichprovides a length scale of the interferometer. Then we obtain L₂=0.417ƒ,dθ₂/dθ₁=1.828, and L₂dθ₂=1.841L₁dθ₁. Thus, L₂dθ₂−L₁dθ₁=0.841L₁dθ₁.

Example 2

As another example the parameters are chosen as ƒ₁=ƒ₂=ƒ, X₁=ƒ, and L₁=ƒ.This yields L₂=0.914ƒ, dθ₂/d₁=1.656, and L₂dθ₂=1.514 L₁dθ₁. Thus,L₂dθ₂−L₁dθ₁=0.514 L₁dθ₁.

These examples demonstrate that it is possible for the phase to retainangular sensitivity even when the two beams A1 and B1 are merged into asingle wavefront C, whose cross-section is much narrower than thedistance L₁ between A₁ and B₁. The incident wavefront C is splitinternally to create the two beams A₁ and B₁. Beams A₂ and B₂ set up asystem of spatial fringes in the vicinity of mirror M₉, which picks upthe phase at its location and recombines the beams. The results ofExamples 1 and 2 indicate that the phase angular sensitivity L₂dθ₂−L₁dθ₁varies with the offset X₁ between the incident beam A₁ and the opticalaxis E₁. This is due to the variation of the spatial frequency of thefringes with the angle of incidence. Thus, a different angularsensitivity is obtained by moving mirror M₉ to a different location.

Mirrors M₈ and M₉ are replaced by the system of mirrors M₁₀, M₁₁, M₁₂,M₁₃ and M₁₄, as shown in FIG. 5, to create an inverted shearconfiguration. A photodetector or CCD array can be placed at anylocation in the observation plane to pick up the phase. Alternatively,mirror M₉ can be used to recombine the beams and provide a temporalsignal. The main distinction between the configurations of FIGS. 3 and 5is that the path length angular sensitivity of FIG. 3 is Ldθ, while thatof FIG. 5 is L₂dθ₂−L₁dθ₁. Nevertheless, the wavefronts A₁ B₁ of FIG. 5are merged into C, whereas the wavefronts A and B in FIG. 3 are notmerged. This invention teaches a method of providing phase sensitivityto changes in the angle of incidence of a single wavefront with areduced cross-section, which is split internally to create two beamsthat are separated by a distance, which is larger than the cross-sectionof either beam, utilizing a system of two curved mirrors whose axes aremisaligned. The design of FIG. 5 provides a compact configuration, whichmakes it possible to turn the whole interferometer to scan the angle ofincidence.

Approach

The perception of an image is determined by its irradiance regardless ofthe field distribution on the object and regardless of whether theobject is illuminated with coherent or incoherent radiation. It would bemore advantageous to synthesize the object from the angular spectrum ofits irradiance distribution, i.e. by measuring the magnitude ofirradiance or intensity as a function of the angle of diffraction. Thesynthetic aperture technique does not do that.

The goal is to produce an image as close to a direct image as possible,i.e. to display the image in real space and time. A direct image is theresult of the superposition and overlap of waves. The creation of thewaves may require some computation but the goal is to simplify thealgorithm and minimize the computations.

The proposed concept is a new imaging technique, which can be applied tocoherent as well as incoherent imaging. When the object is illuminatedwith a broad source or if it is self-luminous, then it can be modeled asa collection of quasimonochromatic point sources, which are incoherent,i.e. do not interfere with each other. However, each point source iscapable of setting up its own interference pattern. If the object is faraway, such as space objects or stars in the sky, then the radiation canbe modeled as a superposition of incoherent collimated waves travelingin slightly different directions. A distant object may be illuminated bya continuous wave laser or a pulsed laser having a finite coherencelength for the purpose of imaging.

It is customary to study the effects of radiation and propagation fortwo limiting cases, namely totally coherent or totally incoherent. Mostcases fall between these two extremes. Since the concept can be used forboth cases with a modification, both cases are illustrated and theirsimilarities and differences pointed out. An effort is made todistinguish between the two types of illumination and keep them separatebut the discussion is entwined.

Diffraction, in the strict sense, occurs only under spatially coherentillumination. The diffracted orders are plane waves, which represent theFourier components of the complex electric field on the object. Eventhough the diffracted waves are modeled as planar, there is actually afinite width associated with them corresponding to the width of theobject. The object and its image are assumed to be small compared toaxial propagation distances, according to Fresnel diffraction theory inorder to satisfy the paraxial approximation. After certain propagationdistance the waves clear each other, i.e. they do not overlap anymore.This is the onset of the far field or the Fraunhofer regime. If theoptical system, such as synthetic aperture is placed in the far field,then the pinholes intercept two conjugate waves, the ones that aretraveling at angles +θ and −θ off the optical axis. The interference ofthe waves diffracted from the pinholes would then yield the ratio of theamplitudes of the waves and their phase difference. By changing thedistance between the pinholes, the diffracted field could be analyzedtwo conjugate waves at a time. If the pinholes were placed closer to theobject before the far field, then each pinhole would intercept more thanone diffracted wave from the object, and the analysis of the diffractionbecomes more complicated.

For incoherent radiation there is no far field per say. The radiationfrom stars and distant objects overlap everywhere in space because eachpoint on the object radiates in all directions. The radiation can bemodeled as a superposition of collimated plane waves incident along thedirection θ corresponding to the angular position of the star. Unlikethe coherent case, these waves cannot be separated no matter how faraway the pinholes are from the object. This faintly resembles the nearfield or Fresnel zone of a coherent object, at least in concept, in thateach pinhole must bear the contribution of more than one plane waveemanating from the object. The total contribution from all theincoherent waves at the location of the pinhole is represented by thecoherence function, which is given by the van Cittert-Zemike theorem.

The theorem states that the Fourier Transform of the irradiancedistribution of the object is given by the coherence function in theplane of the aperture. Thus, there is a Fourier transform relationshipbetween the coordinates in the object plane and the pupil plane.According to Fourier Transform theory, when a function ƒ of coordinatex, i.e. ƒ(x) is transformed it yields a function F(s) where thecoordinate s represents spatial frequency in this case. The coordinate scan be replaced with any other appropriate variable, which isproportional to s. For example, in the case of coherent illumination,the off-axis distance in the focal plane or the lateral distance on ascreen in the far field is replaced by the angle of diffraction, θ,which is related to the spatial frequency s by s=sin(θ)/λ, where λ isthe wavelength of the radiation. The proportionality results under thesmall angle approximation concomitant with paraxial propagation. Thus,the function F(s) becomes F(θ) merely through a substitution ofvariable. This concept applies to incoherent radiation as well, eventhough there is no diffraction from an incoherent object. The separationbetween apertures in the pupil plane Δx and Δy, which are normallyvaried to map the coherence function, can be replaced by the angles ofpropagation, θ_(x) and θ_(y), which are proportional to Δx and Δy,respectively. Thus, a measurement of the intensity vs. angleI(θ_(x),θ_(y)) yields the angular spectrum, which is related to theFourier Transform of the irradiance I(Δx,Δy) through a scaling of thecoordinates. The problem is that the synthetic aperture technique doesnot yield a direct measure of I(θ_(x),θ_(y)), i.e. intensity vs. angleof propagation of the waves from the object toward the pupil plane.Rather, it yields a measure of the angle subtended by the diameter ofthe object looking back from the pupil plane toward the object.

Proposed Concept

It would be advantageous to measure the intensity of the radiation as afunction of the angle of propagation. It is worth noting that this isNOT equivalent to measuring the magnitude of the diffracted field, i.e.the Fourier Transform of the electric field in the far field for acoherently illuminated object. The reason is that while the intensity orirradiance is the square of the electric field at any point in space andtime, it is not so in the frequency domain. Since the intensity is theproduct of the electric field with its complex conjugate, then theFourier Transform of the intensity is the convolution of the FourierTransform of the electric field with that of its complex conjugate, orequivalently, it is the autocorrelation of the Fourier Transform of theelectric field in the frequency domain. This involves the phases of theFourier components of the electric field. Thus, the idea is tosynthesize the irradiance of an object, i.e. construct a direct image byadding the Fourier components of the irradiance rather than those of theelectric field. This would be the case for either coherent or incoherentobjects, except that it is done differently for both cases.

The intensity is the magnitude of the electric field squared in thespatial or temporal domain, but the Fourier component of the intensityat a certain spatial or temporal frequency is NOT equal to the square ofthe magnitude of the Fourier component of the electric field at thatfrequency. Thus, the Fourier components of the intensity cannot beobtained with a lens. The lens separates the different Fouriercomponents of the electric field, i.e. the diffracted orders of acoherent object, in its focal plane and discards the phase information.Similarly, it separates the different collimated waves incident from anincoherent object. This does not yield the Fourier Transform ofintensity for coherent illumination because the phase of the field islost at the focal plane. For incoherent light, focal plane measurementsdo yield a measure of I(θ_(x),θ_(y)) for distant objects, i.e. theangular spectrum of the irradiance but the measurement is verysusceptible to atmospheric aberrations and optical surfaceimperfections. Further, improving the angular resolution requiresincreasing the size of the lens, which becomes impractical andprohibitively expensive beyond a certain limit. It is desired toseparate the image plane from the focal plane, i.e. producenon-localized fringes. Thus, the focal plane of the lens loses itsusefulness. For this reason the lens can be eliminated.

The synthetic aperture technique does not yield the angular spectrum ofthe irradiance. For distant incoherent objects, the angular spectrum ofthe irradiance is more useful than the Fourier Transform. For coherentradiation not enough phase information is provided by the syntheticaperture technique to permit construction of the field or intensityprofile, which explains why the synthetic aperture technique is almostexclusively used in conjunction with incoherent radiation. The syntheticaperture technique does not measure the intensity profile vs. angle ofpropagation. For this reason, a technique is proposed to measure theintensity directly as a function of the angle of propagation, i.e.I(θ_(x),θ_(y)). Thus, the proposed device is an angular sensor. The taskof imaging becomes a matter of sensing and plotting the number ofphotons incident along a certain direction (θ_(x),θ_(y)). The imagingtechnique consists of synthesizing the irradiance pattern directly fromits angular components. The proposed technique is an alternative to thesynthetic aperture technique.

Amplitude Splitting and Wave Splitting Interferometers

All interferometers can be classified as either amplitude or wavesplitting, which are described in textbooks. Amplitude splittinginterferometers, such as Michelson's or Mach-Zehnder split a wavefrontinto two beams using a partially reflecting and transmitting mirror andthen recombine the beams after traveling along different paths.Michelson's interferometer recombines each ray of the wavefront withitself. For this reason it is used for temporal correlation and spectralimaging. The Mach Zehnder can be of the shearing type, i.e. recombinesone portion of the wavefront with another portion shifted laterally; orinverting, i.e. recombines the left half of the wavefront with the righthalf and vice versa. Wavefront splitting interferometers sample thewavefront at two different points, such as the stellar interferometer,which is based on Young's two slit experiment. Two-dimensional spatialimaging has conventionally been done using wavefront splitting. However,amplitude splitting can also be used for spatial imaging in conjunctionwith multiple wavefronts.

It is desired to synthesize the irradiance of an object from its Fourieror angular components. But the Fourier components of the irradiancefunction are not readily available. It requires some level ofcomputation. The goal is to keep the algorithm as simple as possible.Furthermore, it is desired to display the image in real space and time.This requires the overlap of waves, which represent all the Fouriercomponents of the irradiance function. By contrast, the syntheticaperture technique does not provide a real space and time image becausethe waves are sampled only two at a time. The other waves are notavailable simultaneously and the image can be constructed onlycomputationally.

A lens separates the waves in the focal plane and recombines them in theimage plane while maintaining the appropriate angle and phaserelationships. Coherent image formation by a lens, according to ErnstAbbe's theory, which was proposed in 1873, consists of overlapping thewaves in the image plane at the same angles at which they originallydiffracted from the object to recreate the spatial frequencies of theelectric field. The angles are scaled proportionately in the case ofmagnification. The overlapping waves interfere to recreate the electricfield distribution.

There are two ways to reconstruct a real spatial image: Either

1) Overlap the waves to reconstruct the image by interference byattempting to recreate the electric filed conditions that once existedin the object plane. Or,

2) Create waves which represent the Fourier components of the irradianceby whatever means, and just add them without interference.

The first method represents Ernst Abbe's theory. The second technique isthe proposed concept. If the correct Fourier components of the intensityprofile were somehow guessed or computed including their amplitudes,phases and spatial frequencies, and sinusoidal waves were createdrepresenting this information, then the irradiance of the object couldbe reconstructed simply by the superposition of those waves without anyinterference. The task is to create the intensity waves.

If the spatial distribution of the electric field in the object plane isrepresented by E(x,y) where the underline denotes a complex quantityhaving a phase, then the spatial distribution of intensity or irradianceof the object, which is a real positive function, is given byI(x,y)=|E(x,y)|². The Fourier Transform of the electric field, which isalso a complex quantity, is denoted by E(ƒ_(x),ƒ_(y)) where ƒ_(x) andƒ_(y) are the spatial frequencies in the x and y directions,respectively. Similarly, The Fourier Transform of the intensity can beexpressed as I(ƒ_(x),ƒ_(y)), which is a complex quantity having a phaseassociated with the particular Fourier component of intensity. It isimportant to note that I(ƒ_(x),ƒ_(y)) is NOT equal to |E(ƒ_(x))|². Thus,the quantity I(ƒ_(x),ƒ_(y)) cannot be obtained from the focal plane ofthe lens because Fraunhofer diffraction yields only |E(ƒ_(x),ƒ_(y))|².I(ƒ_(x),ƒ_(y)) could however, be computed from |E(ƒ_(x),ƒ_(y))| if thephase of E(ƒ_(x),ƒ_(y)) were known. However, that phase is discarded andcannot be retrieved from focal plane measurements. Given that theintensity is the product of the electric field with its complexconjugate in the spatial domain, the Fourier Transform of the intensityis the autocorrelation of the Fourier Transform of the electric field inthe spatial frequency domain. Computing the autocorrelation entailsshifting a waveform relative to itself and multiplying the complexvalues of the electric field at different frequencies and summing it upover all frequencies. Thus, knowledge of the phases of E(ƒ_(x),ƒ_(y)) atall frequencies is necessary in order to perform the autocorrelation.The synthetic aperture technique yields the phase difference ofE(ƒ_(x),ƒ_(y)) for the coherent case but only for a conjugate pair offrequencies +ƒ and −ƒ, which are intercepted by the instantaneouslocations of the pinholes. No phase relationships among non-conjugateorders are given by the synthetic aperture technique to permitcomputation of the intensity spectrum or the inverse Fourier Transform.For this reason, an alternative interferometric imaging technique isproposed.

The light incident from an object consists of collimated waves. In thecase of coherent illumination the collimated waves are diffracted planewaves. In the incoherent case the waves originate fromquasimonochromatic points on a distant object. The waves travel inslightly different directions off-axis. It is desired to measure theintensity of each wave as a function of the angle of propagation. Theangle of propagation is proportional to a spatial frequency component ofthe object. Thus, if the angle of propagation is measured, i.e. if thenumber of photons that are traveling in a particular direction is knownthen the irradiance of the object could be synthesized from its angularspectrum. There is no optical device, which performs such a functioncurrently other than the lens. In the interest of improving the spatialresolution, it is desired to replace the lens with planar opticalsurfaces, which do not require high fabrication tolerances. The proposedconcept provides an interferometric design which achieves suchfunctionality lenslessly.

Any function can be synthesized from its Fourier components. Inparticular a real function, such as the irradiance of an object, has asymmetrical transform, i.e. the conjugate waves traveling above andbelow the optical axis are in fact hermitian conjugates having equalamplitudes but opposite phases. Thus, the Fourier Transform of theintensity function need only be computed for positive frequencies.However, this requires knowledge of the phases of the electric field forall frequencies to compute the autocorrelation.

Coherent waves from finite objects are separated in the focal plane of alens but made to interfere in the image plane. On the other hand, forincoherent distant objects the focal and image planes merge and thewaves are separated. No interference occurs in the focal plane. Thereconstruction of the image is due to multiple beam interference, whichobtains for coherent as well as incoherent radiation because the wavestravel equal optical path lengths between the object and image planes.This is analogous to the phenomenon that gives rise to Finesse in thetemporal domain. Successive reflections from the two opposing mirrors ofa Fabry-Perot interferometer result in multiple beam interference, whichexhibits peaked reflectance compared to the sinusoidal fringes obtainedwith two-beam interferometers such as Michelson or Mach-Zehnder. Thus,the contributions of all the sub-apertures in the spatial domain areanalogous to the multiple reflections in the temporal domain, exceptthat the path length difference is zero and the free spectral range isinfinite. This explains why a lens produces a single Airy shaped pointspread function, whereas the Fabry-Perot has a periodic transferfunction.

The angle of propagation of a wave, as it leaves the object, representsa particular spatial frequency component. In the case of coherentradiation the intensity of the diffracted wave represents the magnitudeof the electric field squared, not the irradiance of the object. Forincoherent radiation the intensity of the wave represents the irradianceI(θ_(x),θ_(y)). A real function can be synthesized if sinusoidal wavesof the appropriate amplitude, frequency and phase are superposed in aplane or region of space, such that they add algebraically. This is thedefinition of Fourier synthesis. The waves do not have to representelectric fields or interfere with each other. In fact, it would bepreferable if the waves were incoherent so that they just add. Thus, thetask of reconstructing the image becomes that of creating overlappingsinusoidal patterns whose amplitudes and phases correspond to theFourier components of the irradiance pattern. The sinusoidal waves arecreated with the use of amplitude splitting interferometer.

In a wavefront splitting interferometer the spatial frequency of thesinusoidal fringes produced by the diffraction from the holes has norelevance to any spatial frequency of the object. The fringes are merelya carrier and their spatial frequency is determined by the distancebetween the holes Δx and Δy in the pupil plane. In the syntheticaperture technique the coherence is obtained from measurement of thevisibility of the fringes rather than their spatial frequency. Thefringes corresponding to different points on the object have the samefrequency but exhibit a phase shift. When the path length differencebetween the rays from two distant points reaching the two slits variesby λ/2 the two fringe patterns are displaced by half a fringe and thevisibility vanishes if the two point sources are equally bright. This isan important issue which limits the dynamic range of the syntheticaperture technique because of the dependence of the visibility onintensity. This introduces an error in the measurement of phase. Whentwo or more sinusoidal waves with 100% modulation having the samefrequency are added, the resulting wave is sinusoidal of the samefrequency but with a visibility less than one. The visibility of thecombined wave depends not only on the phase differences among theconstituent waves but also on their relative amplitudes. This limits theability of interferometric telescopes to resolve a faint star in thevicinity of a bright star. The resulting visibility measurement dependsnot only on the positions of the stars but also on their relativebrightness. The coherence function being the Fourier Transform of theirradiance of the object depends on the brightness distribution of thesource. The Fourier Transform of a bright source eclipses that of anearby faint source.

By contrast, the proposed technique is not a wavefront splittinginterferometer and does not use sub-apertures. It measures the intensityvs. angle I(θ_(x),θ_(y)) information directly. It is onlyposition-dependent and completely intensity-independent. Each distantpoint source creates a sinusoidal fringe pattern with a differentspatial frequency, which serves as a marker to identify that particularpoint source. The amplitude of the fringe is an indication of thebrightness of the source. This is in contrast to the synthetic aperturetechnique, which represents all point sources by sinusoidal fringes ofthe same frequency but different phases. The proposed technique isunaffected by uneven irradiance distributions on the object. A starlocated at position (θ_(x),θ_(y)) is measured directly to yield thenumber of photons I(θ_(x),θ_(y)) incident along that direction. Themeasurement is not affected by the presence of another star. Theproposed technique measures ‘horizontal’ quantities, i.e. spatialfrequencies rather than ‘vertical’ quantities such as visibility, whichis related to amplitude and phase. Thus, the proposed technique is moreadvantageous than the synthetic aperture technique by virtue of thefrequency measurement, which is more robust than amplitude or phasemeasurement. The concept of visibility is gone. There is no visibilityto measure. The addition of sinusoidal waves of slightly varyingfrequencies does not yield a combined sinusoidal wave with a visibility.The angular spectrum of the incident radiation is measured. Since eachpoint source is represented by a different spatial frequency, a Fouriertransformation of the superposed waves yields the angular spectrum. Eachmeasured spatial frequency corresponds to a certain angular position inthe sky.

The proposed technique consists of superposing sinusoidal waves havingamplitudes and phases corresponding to the Fourier components of theirradiance function and taking the spatial Fourier Transform of thecombined wave. The angular resolution is proportional to the frequencyresolution. Thus, the proposed technique creates a direct image butinvolves a computational step. All the incident waves are interceptedsimultaneously. The superposition of all the interferograms reconstructsthe irradiance distribution in real space and fairly short time. Thereis no need to move any mirrors or apertures. By contrast, the syntheticaperture technique must displace the apertures and involves taking aninverse Fourier Transform to compute the irradiance from the coherencefunction. The synthetic aperture technique does not display an image inreal space and time.

A common feature of all computational techniques is that the data issampled within a finite region of space using the Discrete FourierTransform (DFT). The frequency resolution is inversely proportional tothe total sampling interval. Thus, the system has a finite angularresolution. If the position of a certain star corresponds to a spatialfrequency, which is not a multiple integer of the frequency resolution,then that would lead to leakage, i.e. the intensity from that star leaksinto adjacent frequencies, which smears the image. This can be correctedby increasing the number of data points. Thus, the proposed techniquehas the advantage of immunity to uneven irradiance distributions andhigh dynamic range. It can image a very faint star adjacent to a brightstar by virtue of the robustness of the frequency measurement.

The sinusoidal patterns corresponding to the Fourier components of theirradiance profile are formed using standing waves. Each wave diffractedfrom a coherent object or traveling along a certain direction from adistant incoherent point source is capable of setting up aninterferogram consisting of a sinusoidal fringe pattern, when split andredirected to interfere with itself using an amplitude splittinginterferometer. There is no interference among different waves. Eachwave interferes only with itself to set up a standing wave with acertain spatial frequency that identifies that particular wave uniquely.Since the radiation is redirected using planar mirrors the beammaintains its collimation. The angle of interference can be controlledby orienting the mirrors accordingly. This concept is illustrated inFIG. 6 using a simple Mach-Zehnder configuration. Collimated radiationincident from the object is partially reflected from planar mirror M₁₅.The transmitted and reflected beams are further reflected from planarmirrors M₁₆ and M₁₇, respectively, and intersect at half angle α on theobservation plane S. The spatial frequency of the sinusoidal fringes isgiven by 2(sinα)/λ. Each wave propagating at angle θ sets up a distinctfringe pattern with a different spatial frequency. This provides amarker to identify each incident wave. The fringes are captured with aCCD pixel detector array. The fringes are two-dimensional correspondingto the angles θ_(x) and θ_(y). FIG. 6 is a one-dimensional sketch, whichillustrates the concept. The angle α is related to the angle θ and canbe adjusted by changing the orientations of the mirrors. In order forthe fringe pattern to synthesize the object irradiance the amplitudes,frequencies and phases of the sinusoidal fringes must match those of theFourier components of the object. Ideally, the angle α is made equal tothe angle θ to reproduce the object with unit magnification. If theangle α matches the angle θ for one wave, then it matches it for everywave because the planar mirrors do not perturb the relative angles amongthe waves. The spatial frequency of the fringes can be adjusted bytilting the mirrors appropriately. The amplitudes of the fringes arederived from the irradiance of the object. Thus, the image can bereconstructed in real space using different algorithms for the coherentand incoherent cases. The proposed imaging concept is a full aperturesystem.

Algorithms

For coherent radiation the diffracted wave represents the electricfield. Thus, the amplitude of the fringes corresponds to|E(θ_(x),θ_(y))|². Therefore, the algorithm starts by taking the FourierTransform of the fringes, then the square root of it to yield|E(θ_(x),θ_(y))|. The Fourier Transform of the intensity I(θ_(x),θ_(y))cannot be calculated unless the phase of E(θ_(x),θ_(y)) is known. Butthe phase of E(θ_(x),θ_(y)) cannot be measured. Therefore, an arbitraryphase φ(θ_(x),θ_(y)) is assigned to |E(θ_(x),θ_(y))|, which permits thecomputation of I(θ_(x),θ_(y)) through the autocorrelation of|E(θ_(x),θ_(y))|e^(iφ(θx,θy)). The resulting I(θ_(x),θ_(y)) is comparedto a picture of the object taken with a lens whose aperture is smallerthan that of the proposed imaging system. The process of assigningphases is repeated iteratively until the difference between thecalculated and measured intensity profiles converges within a certainerror limit set by an appropriate criterion.

It is worth noting that the proposed method of synthesizing theintensity by autocorrelating the electric field is advantageous becauseit obviates the need to measure the high frequency components. Since thespectrum of the intensity is the autocorrelation of the electric fieldin the frequency domain, all the components of the electric field forall frequencies contribute to every component of intensity. Therefore,the proposed technique yields a good approximation of the high frequencycomponents of intensity from the measurement of the low frequencycomponents of the electric field.

The main burden of conventional imaging systems is capturing the highfrequency components. For this reason imaging systems strive for largeraperture. This is also true for the proposed imaging system because theangular resolution is inversely proportional to the size of theaperture. However, the burden shifts from capturing the high frequencycomponents to guessing the correct phases of the electric field at lowfrequencies. A fairly good estimate of the high frequency components ofthe intensity can be obtained from the measurement of the lowfrequencies of the electric filed because they contribute to theautocorrelation. This is in contrast with conventional system designwhere the high frequency components can only be inferred from the highfrequency measurements, thereby necessitating an increase in aperture.

In conclusion, if the optical system is oriented correctly relative tothe object then the incident waves set up interferograms whose spatialfrequencies are meaningful Fourier or angular components of theintensity profile of the object. The idea is that each wave interferesonly with itself. The amplitude splitting interferometer can beconfigured to provide enough phase delay among the waves to offset anycoherent effects. The assignment of phases to the components of theelectric field permits calculation of the amplitudes and phases of theFourier components of intensity, which are then compared to measurementsobtained with smaller lensed to systems, i.e. a cheap camera, untiliterative convergence is obtained.

For incoherent radiation the algorithm is simplified significantly. Theamplitude of the fringes is equal to the angular spectrum of theirradiance of the distant object. Thus, a simple Fourier transformationof the fringes yields I(θ_(x),θ_(y)).

Spectral Imaging

Interferometry inherently depends on the wavelength of light. In fact,amplitude splitting interferometers, most notably Michelson's is usedextensively for spectral imaging, which forms the basis of FourierTransform spectroscopy (FTS). The algorithm used to extract spectralinformation is one of the simplest. It consists of taking the FourierTransform of the output of the aligned single wavefront interferometervs. mirror scan distance to yield the power spectrum of the incidentradiation. Tilting of the split wavefronts to create spatial fringes, asshown in FIG. 6, permits spatial imaging. The algorithm for extractionof the spatial image is as simple as that of the spectral image. Thephase of the fringes is adjusted by displacing one of the mirrors M₁₆ orM₁₇ parallel to itself. Two extreme cases have been studied in theanalysis of the proposed concept, namely total coherence and totalincoherence. The fringes can result from variations either in angle orwavelength. One point source emitting at two wavelengths can be confusedfor two quasimonochromatic point sources emitting at the same averagewavelength. The spectral resolution can be estimated from the angularresolution. For small angles the spatial frequency of the fringes ƒ_(s)depends on the ratio of angle to wavelength, θ/λ. Thus, the wavelengthresolution Δλ is given by Δθ/ƒ_(s), where Δθ is the angular resolutiongiven by λ/α, where ‘α’ is the size of the aperture.

Example 3

An aperture of 1 meter provides an angular resolution of 5×10⁻⁷ radianat a wavelength of 500 nanometers. The corresponding spectral resolutionis 10⁻¹¹ meter or 0.1 Angstrom for a fringe period of about 20 μmcompatible with current CCD pixel fabrication technology. This couldaccommodate 100 channels per nanometer of bandwidth, which is adequatefor multi-spectral and hyper-spectral imaging applications.

Conclusion

An important aspect of imaging systems is spatial and spectralresolution. It is desired to widen the aperture to improve the spatialresolution. A lens has the advantage that it performs the FourierTransform optically and requires minimal signal processing. However,increasing the size of the lens beyond a certain limit is impracticaland prohibitively expensive. A concept is provided for a full apertureimaging system utilizing planar reflective optics, which can be scaledto large apertures. The concept uses simple Fourier analysis and analgorithm to compute the angular or Fourier components of the irradiancedistribution. The concept has the advantage that it produces an image inreal space and does not require the fabrication or use of high qualityoptics. It is also more tolerant to atmospheric aberrations. Theproposed concept can be used to image coherent as well as distantincoherent objects. It can be deployed on the ground as well as inspace. It can be used in conjunction with active (laser) illumination,or passive (natural) illumination of objects. The proposed conceptprovides two-dimensional spatial imaging as well as spectral imaging.The concept has the potential to achieve high spatial and spectralresolution. The proposed technique is alternative to the pupil planesynthetic aperture technique.

Having thus described several aspects of at least one embodiment of thisinvention, it is to be appreciated various alterations, modifications,and improvements will readily occur to those skilled in the art. Suchalterations, modifications, and improvements are intended to be part ofthis disclosure, and are intended to be within the spirit and scope ofthe invention. Accordingly, the foregoing description and drawings areby way of example only.

1. A method of imaging an object comprising: Receiving electromagneticradiation comprising wavefronts propagating at different angles,Splitting the amplitudes of the wavefronts, Interfering each wavefrontwith itself to create non-Fizeau fringes having different spatialfrequencies, Combining said non-Fizeau fringes to form an image of theobject.
 2. An interferometer for imaging an object comprising: A meansfor splitting the amplitudes of incoming wavefronts propagating atdifferent angles, A means for interfering each wavefront with itself tocreate non-Fizeau fringes having different spatial frequencies, A meansfor combining said non-Fizeau fringes to form an image of the object.